The band took the stage to perform "Everlong," which as he explained during his closing remarks from the Late Show desk, is a song that has immense personal significance to Letterman — and already had a spot in the show's history.

"Let me tell you a little something about Foo Fighters," Letterman said at the end of a long list of thank-yous. "Fifteen years ago I have open-heart surgery. These people save my life, and I'm out not doing the show for like five or six weeks. I'm in talking with [producer] Sheila Rogers, and she says, 'On your first show back, is there anything special you would like musically?' I said, 'Well, yeah.' We got to talking and we said, 'What about Foo Fighters?' They had been on, I think at that time, four or five times. She said, 'I'll ask them, but it could be a problem.'"

As Letterman went on to explain, he'd leaned on "Everlong" for emotional support and inspiration while recuperating from surgery — something that apparently touched the members of the band enough to help inspire a moving (and seemingly rather expensive) gesture.

"I said, 'Well, just ask them. And also, there's a song of theirs I'd really like to hear — it's special to me, it's been meaningful to my heart recovery. It would be just great.' Two days later, she calls back and says, 'They're on tour in South America.' I said, 'Oh, darn.' She said, 'That's all right — they canceled the tour. They're coming back to do the show, and they're doing the song you asked for.' Ladies and gentlemen, happily ever since, we've been joined at the hip. God bless you, gentlemen, and thank you very much."

Check out the Foo Fighters' Late Show farewell above — and below, you can watch Sting make an appearance on last night's The Late Late Show with James Corden, for a video that found Sting and Corden standing outside the Ed Sullivan Theater singing the Police classic "Every Breath You Take."

Watch Sting and James Corden Sing "Every Breath You Take" in Tribute to David Letterman